Square-free integer
A square-free integer is a number which is not divisible by any square numbers other than 1. In other words, each prime number that appears in its prime factorization appears exactly once.
For example, <math>6 = 2\times3</math> is square-free. However, <math>27 = 3^3</math> is not square-free: it is divisible by <math>9 = 3^2</math>, and the power of <math>3 </math> in the prime factorization is to a power larger than one.
Möbius function[change]
The Möbius function is a function which takes in natural numbers and is usually written as <math>\mu(n)</math>. The value of <math>\mu(n)</math> depends on whether or not <math>n</math> is square-free. Specifically,
- <math>\mu(n) = \begin{cases}
1, &\text{if }n\text{ is square-free and has an even number of prime factors}, \\ -1, &\text{if }n\text{ is square-free and has an odd number of prime factors} \\ 0, &\text{if }n\text{ is not square-free}
\end{cases} </math>
For example, <math>6 = 2\times 3</math> is square-free with an 2 prime factors, so <math>\mu(6) = 1</math>. Since <math>27 = 3^3</math> is not square-free, then <math>\mu(27) = 0</math>.
Since <math>5</math> is prime, it is its own prime decomposition. That is, the prime factorization of <math>5</math> has 1 prime factor, so <math>\mu(5) = -1</math>.